how to solve this question without long calculation

Here is your answer:-
LHS
1−cotθtanθ​+1−tanθcotθ​
=1−tanθ1​tanθ​+1−tanθtanθ1​​

=tanθtanθ−1​tanθ​+tanθ(1−tanθ)1​

=tanθ−1tan2θ​+tanθ(1−tanθ)1​

=tanθ−1tan2θ​−tanθ(tanθ−1)1​

=tanθ(tanθ−1)tan3θ−1​

=tanθ(tanθ−1)(tanθ−1)(tan2θ+tanθ+1)​

=tanθtan2θ+tanθ+1​

=tanθtan2θ​+tanθtanθ​+tanθ1​

=tanθ+1+cotθ

=1+cosθsinθ​+sinθcosθ​

=1+sinθcosθsin2θ+cos2θ​

=1+sinθcosθ1​

=1+secθcosecθ = RHS

Hence, Option (a) is corrrect